I use Opera. It is quick - and it remembers windows you have open from session to session. It will open new windows quietly in the background. It accepts cookies and such. I like it a lot.
It can cope with huge numbers of windows open - though after a while you cannot see what they are, as it loses the writing!
But - it is "buggy" as Craven would say - it is subject to fits of the hissies - sometimes closing itself down for "illegal activities".
I downloaded java with it - and it did odd things to my system, so I uninstalled the java, and things are good.
I do not let it set itself as my default browser, since it is very pushy, and this complicates surfing from emails etc.
I have seen Craven saying that most sites are set up for IE, so it can be less smooth with some sites - eg, earlier versions could not get into Abuzz.
If you choose it - and you can download and trial for free at
www.opera.com - ( I mean, there is a free version - with ads - which is what I use) - I would not bother with opera mail or any such - it is HUGELY aggressive in trying to get you to buy the service, and that is VERY annoying!